The Hoops, who dropped into the competition following their failure to overcome AEK Athens in the Champions League, got off to the perfect start at Marijampole's ARVI Futbolo Arena.
However, the hosts equalised 10 minutes later as a free kick from just inside the Celtic half was poorly defended by Brendan Rodgers's charges.
The Scottish Premiership champions almost retook the lead when James Forrest controlled a cross in the box and let rip with a powerful drive towards the bottom corner of the net.
However, Suduva goalkeeper Ivan Kardum pulled off a stunning reaction save to send the shot wide of target as Celtic had to settle for a draw.
The return leg at Parkhead takes place next Thursday.
SUDUVA (4-1-4-1): Kardum; Svrljuga, Hayduchyk, Jankauskas, Slavickas; Leimonas; Finkler (Offenbacher 45'), Cadjenovic, Verbickas (Matulevicius 90'), Vezevicius (Kasparavicius 60'); Cicilia
CELTIC (3-1-4-2): Gordon; Gamboa (Izaguirre 66'), Simunovic (Lustig 52'), Ajer; Brown; Ntcham, Forrest, McGregor, Tierney; Dembele, Johnston (Griffiths 66')